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This source concerns signed sums and families of subsets. Its published-page scan contains a GDZ archive cover, then printed pages 251–259 on PDF pages 2–10. The mathematical version is the 1965 journal article.

Finite set notation

Write $\mathcal B(S)$ for all subsets of a finite set $S$ and $\mathcal B_n$ when $|S|=n$. In particular, $\mathcal B_0=\{\varnothing\}$. An antichain contains no two distinct comparable members.

A saturated symmetric chain in $\mathcal B_n$ contains one set at each rank $k,k+1,\ldots,n-k$, for some $0\le k\le\lfloor n/2\rfloor$, with successive sets related by inclusion. Its length is its number of members, $n-2k+1$, rather than its number of edges. The source calls these subchains of maximal chains. Empty lists arising in the induction are discarded.

Set $\binom nt=0$ for integers $t$ outside $0\le t\le n$. For $r\ge1$, write

$$B_n(r)= \begin{cases} \displaystyle\binom n{\lfloor(n+r)/2\rfloor},&1\le r\le n+1,\\ 0,&r>n+1. \end{cases}$$

These are the chain-tail counts proved at the remark after Lemma I. The number $q$ of antichains is a nonnegative integer. Its union is empty when $q=0$; the bound by largest rank levels is truncated at $n+1$. All subset-family bounds use $\le$, permitting equality.

Signed sums and discs

For $a_1,\ldots,a_n\in\mathbb C$, the counted objects are sign choices $\varepsilon\in\{-1,1\}^n$ for which $s(\varepsilon)=\sum_i\varepsilon_i a_i$ lies in a region. Distinct choices giving the same complex number are counted separately. The unique empty choice at $n=0$ has sum zero.

A closed unit disc is $\{z:|z-c|\le1\}$; an open unit disc is $\{z:|z-c|<1\}$. The plane theorem uses $|a_i|>1$ and allows a closed disc. The finite scaling consequence uses $|a_i|\ge1$ and requires an open disc. The norm-one closed-disc version is false.

Inputs and coverage

The plane chain proves Lemma I, the chain-count remark, Lemma II, Theorem II, Theorem I, and the open-disc transfer. It includes the binomial identity used by Theorem II and uses only finite induction, finite counting, and elementary Euclidean inner-product and rotation facts. Although the source attributes Lemma II to Erdős 1945, its proof is included locally and is not an unproved imported input.

The later higher-dimensional branch has only stated source claims and proof pointers here. No complete proof of its Lemmas III/IV or Theorem III is claimed.

Bears on. Problem 498, with the explicit disc and multiplicity conventions above.